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Decision-making with Sugeno integrals: DMU vs.
MCDM
Miguel Couceiro, Didier Dubois, Henri Prade, Tamas Waldhauser
To cite this version:
Miguel Couceiro, Didier Dubois, Henri Prade, Tamas Waldhauser. Decision-making with Sugeno
integrals: DMU vs. MCDM. ECAI 2012 - 20th European Conference on Artificial Intelligence., Aug
2012, Montpellier, France. �hal-01093646�
Decision-making with Sugeno integrals: DMU vs. MCDM
Miguel Couceiro
1and Didier Dubois
2and Henri Prade
3and Tam´as Waldhauser
4Abstract. This paper clarifies the connection between multiple cri-teria decision-making and decision under uncertainty in a qualitative setting relying on a finite value scale. While their mathematical for-mulations are very similar, the underlying assumptions differ and the latter problem turns out to be a special case of the former. Sugeno integrals are very general aggregation operations that can represent preference relations between uncertain acts or between multifactorial alternatives where attributes share the same totally ordered domain. This paper proposes a generalized form of the Sugeno integral that can cope with attributes which have distinct domains via the use of qualitative utility functions. In the case of decision under uncertainty, this model corresponds to state-dependent preferences on act conse-quences. Axiomatizations of the corresponding preference function-als are proposed in the cases where uncertainty is represented by pos-sibility measures, by necessity measures, and by general monotonic set-functions, respectively. This is achieved by weakening previously proposed axiom systems for Sugeno integrals.
1
MOTIVATION
Two important chapters of decision theory are decision under uncer-tainty and multicriteria evaluation [4]. Although these two areas have been developed separately, they entertain close relationships. On the one hand, they are not mutually exclusive; in fact, there are works dealing with multicriteria evaluation under uncertainty [29]. On the other hand, the structure of the two problems is very similar, see, e.g., [18, 20]. Decision-making under uncertainty (DMU), after Sav-age [35], relies on viewing a decision (called an act) as a mapping from a set of states of the world to a set of consequences, so that the consequence of an act depends on the circumstances in which it is performed. Uncertainty about the state of the world is represented by a set-function on the set of states, typically a probability measure.
In multicriteria decision-making (MCDM) an alternative is eval-uated in terms of its (more or less attractive) features according to prescribed attributes and the relative importance of such features. Attributes play in MCDM the same role as states of the world in DMU, and this very fact highlights the similarity of alternatives and acts: both can be represented by tuples of ratings (one per state or objects). Moreover, importance coefficients in MCDM play the same role as the uncertainty function in DMU. A major difference between MCDM and DMU is that in the latter there is usually a unique conse-quence set, while in MCDM each attribute possesses its own domain. A similar setting is that of voting, where voters play the same role as attributes in MCDM.
There are several possible frameworks for representing decision
1University of Luxembourg, Luxembourg, email: miguel.couceiro@uni.lu 2IRIT - Universit´e Paul Sabatier, France, email: Dubois@irit.fr
3IRIT - Universit´e Paul Sabatier, France, email: Prade@irit.fr 4University of Szeged, Hungary, email: twaldha@math.u-szeged.hu
problems that range from numerical to qualitative and ordinal. While voting problems are often cast in a purely ordinal setting (leading to the famous impossibility theorem of Arrow), decision under uncer-tainty adopts a numerical setting as it deals mainly with quantities (since its tradition comes from economics) . The situation of MCDM in this respect is less clear: the literature is basically numerical, but many methods are inspired by voting theory; see [5].
In the last 15 years, the paradigm of qualitative decision theory has emerged in Artificial Intelligence in connection with problems such as webpage configuration, recommender systems, or ergonomics (see [17]). In such topics, quantifying preference in very precise terms is difficult but not crucial, as these problems require on-line inputs from humans and must be provided in a rather short period of time. As a consequence, the formal models are either ordinal (like in CP-nets, see [3]) or qualitative, that is, based on finite value scales. This paper is a contribution to evaluation processes in the finite value scale setting for DMU and MCDM. In such a qualitiative setting, the most natural aggregation functions are based on the Sugeno inte-gral. Theoretical foundations for them (in the scope of DMU) have been proposed in the setting of possibility theory [24], and assuming a more general representation of uncertainty [23]. The same aggre-gation functions have been used in [30] in the scope of MCDM, and applied in [32] to ergonomics. In these papers it is assumed that the domains of attributes are the same totally ordered set.
In the current paper, we remove this restriction, and consider an aggregation model based on compositions of Sugeno integrals with qualitative utility functions on attribute domains, we call Sugeno util-ity functionals. We propose an axiomatic approach to these extended preference functionals that enables the representation of preference relations over Cartesian products of, possibly different, finite chains (scales). We consider the cases when importance weights bear on in-dividual attributes (the importance function is then a possibility or a necessity measure), and the general case when importance weights are assigned to groups of attributes, not necessarily singletons. We study this extended Sugeno integral framework in the DMU situa-tion showing it leads to the case of state-dependent preferences on consequences of acts. The new axiomatic system is compared to pre-vious proposals in qualitative DMU: it comes down to deleting or weakening two axioms on the global preference relation.
The paper is organized as follows. Section 2 introduces basic no-tions and terminology, and recalls previous results needed throughout the paper. Our main results are given in Section 3, namely, represen-tation theorems for multicriteria preference relations by Sugeno util-ity functionals. In Section 4, we compare this axiomatic approach to that previously presented in DMU. We show that this new model can account for preference relations that cannot be represented in DMU, i.e., by Sugeno integrals applied to a single utility function. However there is no increase of expressive power in the case of possibility theory. Proofs are omitted due to space limitations.
2
BASIC BACKGROUND
In this section, we recall basic background and present some prelim-inary results needed throughout the paper. For introduction on lattice theory see [33].
2.1
Preliminaries
Throughout this paper, let Y be a finite chain endowed with lattice operations∧ and ∨, and with least and greatest elements 0Y and
1Y, respectively; the subscripts may be omitted when the underlying
lattice is clear from the context; [n] is short for{1, . . . , n} ⊂ N. Given finite chains Xi, i ∈ [n], their Cartesian product X =
∏
i∈[n]Xiconstitutes a bounded distributive lattice by defining
a∧b = (a1∧b1, . . . , an∧bn), and a∨b = (a1∨b1, . . . , an∨bn).
In particular, a ≤ b if and only if ai ≤ bifor every i∈ [n]. For
k ∈ [n] and c ∈ Xk, we use xck to denote the tuple whose i-th
component is c, if i = k, and xi, otherwise.
Let f : X → Y be a function. The range of f is given by ran(f ) ={f(x) : x ∈ X}. Also, f is said to be order-preserving if, for every a, b∈∏i∈[n]Xisuch that a≤ b, we have f(a) ≤ f(b).
A well-known example of an order-preserving function is the median function med : Y3→ Y given by
med(x1, x2, x3) = (x1∧ x2)∨ (x1∧ x3)∨ (x2∧ x3).
2.2
Basic background on polynomial functions and
Sugeno integrals
In this subsection we recall some well-known results concerning polynomial functions that will be needed hereinafter. For further background, we refer the reader to, e.g., [16, 26].
Recall that a (lattice) polynomial function on Y is any map
p : Yn → Y which can be obtained as a composition of the
lat-tice operations∧ and ∨, the projections x 7→ xi and the constant
functions x7→ c, c ∈ Y .
As shown by Goodstein [25], polynomial functions over bounded distributive lattices (in particular, over bounded chains) have very neat normal form representations. For I⊆ [n], let 1Ibe the
charac-teristic vector of I, i.e., the n-tuple in Ynwhose i-th component is 1 if i∈ I, and 0 otherwise.
Theorem 1. A function is a polynomial function if and only if
p(x1, . . . , xn) = ∨ I⊆[n] ( p(1I)∧ ∧ i∈I xi ) . (1)
Equivalently, p : Yn→ Y is a polynomial function if and only if p(x1, . . . , xn) = ∧ I⊆[n] ( p(1[n]\I)∨ ∨ i∈I xi ) .
Remark 1. Observe that, by Theorem 1, every polynomial function p : Yn → Y is uniquely determined by its restriction to {0, 1}n. Also, since every lattice polynomial function is order-preserving, the coefficients in (1) are monotone increasing as well, i.e., p(eI) ≤
p(eJ) whenever I⊆ J. Moreover, a function f : {0, 1}n→ Y can
be extended to a polynomial function over Y if and only if it is order-preserving.
Polynomial functions are known to generalize certain prominent fuzzy integrals, namely, so-called Sugeno integrals. A fuzzy measure on [n] is a mapping µ :P([n]) → Y which is order-preserving (i.e., if A⊆ B ⊆ [n], then µ(A) ≤ µ(B)) and satisfies µ(∅) = 0 and
µ([n]) = 1; such functions qualify to represent uncertainty.
The Sugeno integral associated with the fuzzy measure µ is the function qµ: Yn→ Y defined by qµ(x1, . . . , xn) = ∨ I⊆[n] ( µ(I)∧∧ i∈I xi ) . (2)
For further background see, e.g., [28, 36, 37].
Remark 2. As observed in [30, 31], Sugeno integrals coincide
ex-actly with those polynomial functions q : Yn→ Y which are
idem-potent, that is, which satisfy q(c, . . . , c) = c, for every c ∈ Y . In
fact, by (1) it suffices to verify this identity for c∈ {0, 1}, that is,
q(1[n]) = 1 and q(1∅) = 0.
Remark 3. Note also that the range of a Sugeno integral q : Yn→ Y
is ran(q) = Y . Moreover, by defining µ(I) = q(1I), we get q = qµ.
In the sequel, we shall be particularly interested in the following types of fuzzy measures. A fuzzy measure µ is called a possibility
measure (resp. necessity measure) if for every A, B ⊆ [n], µ(A ∪ B) = µ(A)∨ µ(B) (resp. µ(A ∩ B) = µ(A) ∧ µ(B)).
Remark 4. In the finite setting, a possibility measure is completely
characterized by the value of µ on singletons, namely µ(i), i∈ [n] (called a possibility distribution), since clearly, µ(A) =∨i∈Aµ(i).
Likewise, a necessity measure is completely characterized by the value of µ on sets of the form Ni = [n]\ {i} since clearly,
µ(A) =∧i6∈Aµ(Ni)
Note that if µ is a possibility measure [38] (resp. necessity mea-sure [22]), then qµis a weighted disjunction
∨
i∈Iµ(i)∧ xi(resp.
weighted conjunction µ(I)∧∧i∈Ixi)) for some I⊆ [n] [21] (where
µ(i), a shorthand notation for µ({i}), represents importance of
cri-terion i). The weighted disjunction operation is then permissive (it is enough that one important criterion be satisfied for the result to be high) and the weighted conjunction is demanding (all important criteria must be satisfied).
Polynomial functions and Sugeno integrals have been character-ized by several authors, and in the more general setting of distributive lattices see, e.g., [7, 8, 28].
The following characterization in terms of median decomposabil-ity will be instrumental in this paper. A function p : Yn→ Y is said
to be median decomposable if for every x∈ Yn,
p(x) = med(p(x0k), xk, p(x1k)
)
(k = 1, . . . , n). Theorem 2([6, 31]). Let p : Yn→ Y be a function on an arbitrary
bounded chain Y . Then p is a polynomial function if and only if p is median decomposable.
2.3
Sugeno utility functionals
Let X1, . . . , Xnand Y be finite chains. We denote (with no danger
of ambiguity) the top and bottom elements of X1, . . . , Xnand Y by
1 and 0, respectively.
We say that a mapping ϕi: Xi → Y , i ∈ [n], is a local utility
function if it is order-preserving. It is a qualitative utility function as
mapping on a finite chain. A function f : X→ Y is a Sugeno utility
functional if there is a Sugeno integral q : Yn→ Y and local utility
functions ϕi: Xi→ Y , i ∈ [n], such that
Note that Sugeno utility functionals are order-preserving.
Remark 5. (i) In [13] it was shown that the set of functions obtained
by composing lattice polynomials with local utility functions is the same as the set of Sugeno utility functionals.
(ii) In [13] and [14] a more general setting was considered, where the inner functions ϕi: Xi → Y , i ∈ [n], were only required to
satisfy the so-called “boundary conditions”: for every x∈ Xi,
ϕi(0)≤ ϕi(x)≤ ϕi(1) or ϕi(1)≤ ϕi(x)≤ ϕi(0). (4)
The resulting compositions (3) where q is a polynomial function (resp. Sugeno integral) were referred to as “pseudo-polynomial functions” (resp. “pseudo-Sugeno integrals”). As it turned out, these two notions are in fact equivalent.
(iii) Note that pseudo-polynomial functions are not necessarily order-preserving, and thus they are not necessarily Sugeno utility func-tionals. However, Sugeno utility functionals coincide exactly with those polynomial functions (or, equivalently, pseudo-Sugeno integrals) which are order-preserving, see [13].
Sugeno utility functionals can be axiomatized in complete analogy with polynomial functions by extending the notion of median decom-posability. We say that f : X→ Y is pseudo-median decomposable if for each k∈ [n] there is a local utility function ϕk: Xk→ Y such
that
f (x) = med(f (x0k), ϕk(xk), f (x1k)
)
(5) for every x∈ X.
Theorem 3([13]). A function f : X→ Y a Sugeno utility functional
if and only if f is pseudo-median decomposable.
Remark 6. In [13] and [14] a more general notion of
pseudo-median decomposability was considered where the inner functions
ϕi: Xi → Y , i ∈ [n], were only required to satisfy the boundary
conditions.
Note that once the local utility functions ϕi: Xi → Y (i ∈ [n])
are given, the pseudo-median decomposability formula (5) provides a disjunctive normal form of a polynomial function p0which can be
used to factorize f . To this extent, let b1I denote the characteristic
vector of I ⊆ [n] in X, i.e., b1I ∈ X is the n-tuple whose i-th
component is 1Xiif i∈ I, and 0Xiotherwise.
Theorem 4([14]). If f : X → Y is pseudo-median
decompos-able w.r.t. local utility functions ϕk: Xk → Y (k ∈ [n]), then
f = p0(ϕ1, . . . , ϕn), where the polynomial function p0is given by p0(y1, . . . , yn) = ∨ I⊆[n] ( f(b1I ) ∧∧ i∈I yi ) . (6)
This result naturally asks for a procedure to obtain local utility functions ϕi: Xi → Y (i ∈ [n]) which can be used to factorize
a given Sugeno utility functional f : X → Y into a composition (3). In the more general setting of pseudo-polynomial functions, such procedures were presented in [13] when Y is an arbitrary chain, and in [14] when Y is a finite distributive lattice.
The following result provides a noteworthy axiomatization of Sugeno utility functionals which follows as a corollary of Theorem 19 in [14].
Theorem 5. A function f : X→ Y is a Sugeno utility functional if
and only if it is order-preserving and satisfies f(x0k ) < f (xak) and f (y a k) < f ( y1k ) =⇒ f (xak)≤ f (y a k)
for all x, y∈ X and k ∈ [n], a ∈ Xk.
Let us interpret this result in terms of multicriteria evaluation. Consider alternatives x and y such that xk = yk = a. Then
f(x0
k
)
< f (x) means that down-grading attribute k makes the
cor-responding alternative x0k strictly worse than x. Similarly, f (y) <
f(y1
k
)
means that upgrading attribute k makes the corresponding alternative y1kstrictly better than y. This behavior is due to a
non-compensatory property of qualitative aggregation operators, which here takes the form of pseudo-median decomposibility. Indeed what this property expresses is that the value of x is either x0k, or x
1
k
or xk. In such a situation, given another alternative y such that
yk= xk= a : f(x0k ) < f (x) = med(f (x0k), ϕk(a), f (x1k) ) = ϕk(a)∧ f(x1k)≤ ϕk(a), f(y1k )
> f (y) = med(f (y0k), ϕk(a), f (y1k)
) = ϕk(a)∨ f(y0k)≥ ϕk(a),
and so f (x) ≤ ϕk(a) ≤ f (y). Hence, if maximally
downgrad-ing (resp. upgraddowngrad-ing) attribute k makes the alternative worse (resp. better) it means that its overall rating was not more (resp. not less) that the rating on attribute k. It also means that either attribute k can affect the value of y positively or it can affect it negatively, but not both. We shall further discuss these facts in Section 5.
It is also interesting to comment on Sugeno utility functionals as opposed to Sugeno integrals applied to a single local utility function. First, the role of local utility functions is clearly to embed all the local scales Xi into a single scale Y in order to make the scales
Xicommensurate. In other words, a Sugeno integral (7) cannot be
defined if there is no common scale X such that Xi⊆ X, for every
i∈ [n]. In particular, the situation in decision under uncertainty is
precisely that where Xi= X, for every i∈ [n], that is, the utility of a
consequence resulting from implementing an act does not depend on the state of the world in which the act is implemented. Then it is clear that ϕi= ϕ, for every i∈ [n], namely, a unique utility function is at
work. In this sense, the Sugeno utility functional becomes a simple Sugeno integral of the form
qµ(y1, . . . , yn) = ∨ I⊆[n] ( µ(I)∧∧ i∈I yi ) . (7)
where Y = ϕ(X). This is the case for DMU, where [n] is the set of states of nature, and X is the set of consequences (not necessarily ordered). It is the utility function ϕ that equips X with a total order:
xi ≤ xj ⇐⇒ ϕ(xi) ≤ ϕ(xj). The general case studied here
corresponds to that of DMU but where the local utility functions ϕi:
X → Y are state-dependent; this situation was already considered
in the literature of expected utility theory [34], here adapted to the qualitative setting. Namely, an act is of the form x∈ Xnwhere the consequences xiof the act performed in state i belong to the same
set X, and the evaluation of x is of the form (3), i.e. they are not evaluated in the same way in each state.
3
PREFERENCE RELATIONS REPRESENTED
BY SUGENO UTILITY FUNCTIONALS
In this section we are interested in relations which can be represented by Sugeno utility functionals. In Subsection 3.1 we recall basic no-tions and present preliminary observano-tions pertaining to preference relations. We discuss several axioms of MCDM in Subsection 3.2 and present several equivalences between them. In Subsections 3.3
and 3.4 we present axiomatizations of those preference relations in-duced by possibility and necessity measures, and of more general preference relations represented by Sugeno utility functions.
3.1
Preference relations on Cartesian products
One of the main areas in decision making is the representation of preference relations. A weak order on a set X =∏i∈[n]Xiis a
rela-tion- ⊆ X2that is reflexive, transitive, and complete (∀x, y ∈ X :
x - y or y - x). Like quasi-orders (i.e., reflexive and transitive
relations), weak orders do not necessarily satisfy the antisymmetry
condition:
∀x, y ∈ X : x - y, y - x =⇒ x = y (AS) This fact gives rise to an “indifference” relation which we denote by
∼, and which is defined by y ∼ x if x - y and y - x. Clearly, ∼ is
an equivalence relation. Moreover, the quotient relation- / ∼ satis-fies (AS); in other words,- / ∼ is a complete linear order (chain). For notational ease, we shall denote- / ∼ by ≤.
By a preference relation on X we mean a weak order- which satisfies the Pareto condition:
∀x, y ∈ X : x ≤ y =⇒ x - y. (P) In this section we are interested in modeling preference relations, and in this field two problems arise naturally. The first deals with the representation of such preference relations, while the second deals with the axiomatization of the chosen representation. Concerning the former, the use of aggregation functions has attracted much attention in recent years, for it provides an elegant and powerful formalism to model preference [4, 27] (for general background on aggregation functions, see [28, 1]).
In this approach, a relation- on a set X = ∏i∈[n]Xiis
rep-resented by a so-called global utility function U (i.e., an order-preserving mapping which assigns to each event in X an overall score in a possibly different scale Y ), under the rule: x - y if and only if U (x)≤ U(y). Such a relation is clearly a preference relation. Conversely, if- is a preference relation, then the canonical sur-jection r : X→ X/ ∼, also referred to as the rank function of -, is an order-preserving map from X to X/∼ (linearly ordered by ≤), and we have x- y ⇐⇒ r (x) ≤ r (y). Thus, - is represented by an order-preserving function if and only if it is a preference relation, and in this case- is represented by r.
3.2
Axioms pertaining to preference modelling
In this subsection we recall some properties of relations used in the axiomatic approach discussed in [20, 23]; here, we will adopt the same terminology even if its motivation only makes sense in the realm of decision making under uncertainty. We also introduce some variants, and present connections between them.
First, for x, y ∈ X and A ⊆ [n], let xAy denote the tuple in
X whose i-th component is xiif i∈ A and yiotherwise. 0 and 1
denote the bottom and the top of X respectively.
We consider the following axioms. The optimism axiom
∀x, y ∈ X, ∀A ⊆ [n] : xAy ≺ x =⇒ x - yAx, (OPT) which subsumes5two instances of interest, namely,
∀x ∈ X, ∀A ⊆ [n] : xA0 ≺ x =⇒ x - 0Ax, (OPT0)
∀x, y ∈ X, k ∈ [n] , a ∈ Xk: x0k≺ x a k =⇒ x a k- y a k. (OPT1)
5For (OPT) =⇒ (OPT
1), just take x = xak, y = yk0and A = [n]\ {k}.
Note that under (P) the conclusion of (OPT0) is equivalent to x ∼
0Ax. Similarly, the conclusion of (OPT1) could be replaced by xak∼
0ak. The name optimism is justified considering the case where X =
1 and Y = 0. Then (OPT) reads Ac ≺ [n] implies A % [n] (full
trust in A or Ac, an optimistic approach to uncertainty). Dual to optimism we have the pessimism axiom
∀x, y ∈ X, ∀A ⊆ [n] : xAy x =⇒ x % yAx, (PESS) which subsumes the two dual instances
∀x ∈ X, ∀A ⊆ [n] : xA1 x =⇒ x % 1Ax, (PESS0)
∀x, y ∈ X, k ∈ [n] , a ∈ Xk: x1k xak =⇒ xak% yak. (PESS1)
Again, under (P), the conclusions of (PESS0) and (PESS1) are
equiv-alent to x ∼ 1Ax and xak ∼ 1 a
k, respectively. When X = 0 and
Y = 1, (PESS) reads Ac ∅ implies ∅ % A (full distrust in A or
Ac, a pessimistic approach to uncertainty).
We will also consider the disjunctive and conjunctive axioms
∀y, z ∈ X : y ∨ z ∼ y or y ∨ z ∼ z, (∨)
∀y, z ∈ X : y ∧ z ∼ y or y ∧ z ∼ z. (∧) Moreover, we have the so-called disjunctive dominance and strict
disjunctive dominance
∀x, y, z ∈ X : x % y, x % z =⇒ x % y ∨ z, (DD%)
∀x, y, z ∈ X : x y, x z =⇒ x y ∨ z, (DD ) as well as their dual counterparts, conjunctive dominance and strict
conjunctive dominance,
∀x, y, z ∈ X : y % x, z % x =⇒ y ∧ z % x, (CD%)
∀x, y, z ∈ X : y x, z x =⇒ y ∧ z x. (CD ) Theorem 6. If - is a preference relation, then axioms (OPT), (OPT0), (OPT1), (∨), (DD%) and (DD ) are pairwise equivalent.
Dually, we have the following result which establishes the pair-wise equivalence between the remaining axioms.
Theorem 7. If - is a preference relation, then axioms (PESS), (PESS0), (PESS1), (∧), (CD%) and (CD ) are pairwise equivalent.
3.3
Preference relations induced by possibility and
necessity measures
In this subsection we present some preliminary results towards the axiomatization of preference relations represented by Sugeno utility functionals (see Theorem 10). More precisely, we first obtain an ax-iomatization of relations represented by Sugeno utility functionals associated with possibility measures (weighted disjunction of utility functions).
Theorem 8. A preference relation- satisfies one (or, equivalently,
all) of the axioms in Theorem 6 if and only if there are local utility functions ϕi, i ∈ [n], and a possibility measure µ, such that - is
represented by the Sugeno utility functional f = qµ(ϕ1, . . . , ϕn).
Remark 7. Note that the above theorem does not state that every
Sugeno utility functional representing a preference relation that sat-isfies the conditions of Theorem 6 corresponds to a possibility mea-sure. As an example, consider the case n = 2 with X1 = X2 =
{0, 1} and Y = {0, a, b, 1}, where 0 < a < b < 1. Let us define
local utility functions ϕi: Xi→ Y (i = 1, 2) by
ϕ1(0) = 0, ϕ1(1) = b, ϕ2(0) = a, ϕ2(1) = 1,
and let µ be the fuzzy measure on{1, 2} given by
µ (∅) = 0, µ ({1}) = a, µ ({2}) = b, µ ({1, 2}) = 1.
It is easy to see that µ is not a possibility measure, but the preference relation - on X1 × X2 represented by f := qµ(ϕ1, ϕ2) clearly
satisfies (∨), since (0, 0) ∼ (1, 0) ≺ (0, 1) ∼ (1, 1) . On the other hand, the same relation can be represented by the second projection (x1, x2) 7→ x2 on{0, 1}6, which is in fact a Sugeno integral with
respect to a possibility measure satisfying 0 = µ(∅) = µ({1}) and
µ({2}) = µ({1, 2}) = 1.
Concerning necessity measures, by duality, we have the following characterization of the weighted conjunction of utility functions. Theorem 9. A preference relation- satisfies one (or, equivalently,
all) of the axioms in Theorem 7 if and only if there are local utility functions ϕi, i ∈ [n], and a necessity measure µ, such that - is
represented by the Sugeno utility functional f = qµ(ϕ1, . . . , ϕn).
3.4
Axiomatizations of preference relations
represented by Sugeno utility functionals
Recall that- is a preference relation if and only if - is represented by an order-preserving function valued in some chain (for instance, by its rank function). The following result that draws from Theorem 5 (whose meaning is discussed above) axiomatizes those preference relations represented by general Sugeno utility functionals.
Theorem 10. A preference relation- on X can be represented by a
Sugeno utility functional if and only if
x0k≺ x a k and y a k≺ y 1 k =⇒ x a k- y a k (8)
holds for all x, y∈ X and k ∈ [n], a ∈ Xk.
4
DMU vs. MCDM
In [23], Dubois, Prade and Sabbadin, considered the qualitative set-ting under uncertainty, and axiomatized those preference relations on X = Xnthat can be represented by special (state-independent) Sugeno utility functionals f : X→ Y of the form
f (x) = p(ϕ(x1), . . . , ϕ(xn)), (9)
where p : Yn → Y is a polynomial function (or, equivalently, a
Sugeno integral; see, e.g.,[9, 10]), and ϕ : X→ Y is a utility func-tion. To get it, two additional axioms (more stringent than (DD%) and (CD%)) were considered, namely, the so-called restrictive disjunctive
dominance and restrictive conjunctive dominance:
∀x, y, c ∈ X : x y, x c =⇒ x y ∨ c, (RDD)
∀x, y, c ∈ X : y x, c x =⇒ y ∧ c x, (RCD) where c is a constant tuple.
Theorem 11(In [23]). A preference relation- on X = Xncan be represented by a state-independent Sugeno utility functional (9) if and only if it satisfies (RDD) and (RCD).
6 Since X/ ∼ has two elements, this is essentially the same as the rank
function r : X→ X/ ∼.
Clearly, (9) is a particular form of (3), and thus every preference relation- on X = Xnwhich is representable by (9) is also repre-sentable by a Sugeno utility functional (3). In other words, we have that (RDD) and (RCD) imply condition (8). However, as the follow-ing example shows, the converse is not true.
Example 12. Let X = {1, 2, 3} = Y endowed with the natural ordering of integers, and the consider the preference relation- on
X = X2whose equivalence classes are
[(3, 3)] ={(3, 3), (2, 3)},
[(3, 2)] ={(3, 2), (3, 1), (1, 3), (2, 2), (2, 1)}, [(1, 2)] ={(1, 2), (1, 1)}.
This relation does not satisfy (RDD), e.g., take x = (2, 3), y = (1, 3) and c = (2, 2) (similarly, it does not satisfy (RCD)), and thus it cannot be represented by a Sugeno utility functional (9). How-ever, with q(x1, x2) = (2∧ x1)∨ (2 ∧ x2) ∨ (3 ∧ x1 ∧ x2), and ϕ1 ={(3, 3), (2, 3), (1, 1)} and ϕ2 = {(3, 3), (2, 1), (1, 1)},
we have that - is represented by the Sugeno utility functional
f (x1, x2) = q(ϕ1(x1), ϕ2(x2)).
In the case of preference relations induced by possibility and ne-cessity measures, Dubois, Prade and Sabbadin [24] obtained the fol-lowing axiomatizations.
Theorem 13(In [24]). Let- be a preference relation on X = Xn. Then the following assertions hold.
(i) - satisfies (OPT) and (RDD) if and only if there exist a utility
function ϕ and a possibility measure µ, such that- is represented by the Sugeno utility functional f = qµ(ϕ, . . . , ϕ).
(ii) - satisfies (PESS) and (RCD) if and only if there exist a utility
function ϕ and a necessity measure µ, such that- is represented by the Sugeno utility functional f = qµ(ϕ, . . . , ϕ).
Again, every preference relation which is representable as in (i) or (ii) of Theorem 13, is representable as in Theorems 8 and 9, re-spectively. Surprisingly and unlike the comparison of those models arising from (3) and(9) (where the latter was shown to be strictly subsumed by the former), every preference relation which is repre-sentable as in Theorems 8 and 9 (when X = Xn) is representable as in (i) or (ii) of Theorem 13, respectively.
To see this, suppose that- is representable by a Sugeno utility functional f = qµ(ϕ1, . . . , ϕn), where ϕi: X → Y and µ is a
possibility measure. Note that f (x) = ∨i∈Iµ({i}) ∧ ϕi(xi), for
some I ⊆ [n]. We claim that satisfies (RDD); by Theorem 8, -satisfies (OPT) (or, equivalently, all of the axioms in Theorem 6).
So let x, y, c∈ X such that x y and x c, i.e., f(x) > f(y) and f (x) > f (c). Since Y is a chain, f (x) > f (y)∨ f(c).
As observed, f (x) =∨i∈Iµ({i}) ∧ ϕi(xi), and thus
f (y)∨ f(c) =( ∨ i∈I µ({i}) ∧ ϕi(yi) ) ∨( ∨ i∈I µ({i}) ∧ ϕi(c) ) =∨ i∈I µ({i}) ∧(ϕi(yi)∨ ϕi(c) ) = f (y∨ c). Hence f (x) > f (y∨ c), which shows that x y ∨ c.
Dually, we can show that if- is representable by a Sugeno utility functional f = qµ(ϕ1, . . . , ϕn), where ϕi: X → Y and µ is a
necessity measure, then satisfies (RCD) and, by Theorem 9, -satisfies (OPT) (or, equivalently, all of the axioms in Theorem 7).
In view of Theorem 13, we have just proved the following result which basically states that the DMU and MCDM settings have the same expressive power w.r.t. possibility and necessity measures.
Theorem 14. Let- be a preference relation on X = Xn. Then the
following assertions hold.
(i) - is represented by a Sugeno utility functional f =
qµ(ϕ1, . . . , ϕn) for a possibility measure µ and utility functions
ϕiif and only if there exist a utility function ϕ and a possibility
measure µ0, such that- is represented by f = qµ0(ϕ, . . . , ϕ).
(i) - is represented by a Sugeno utility functional f =
qµ(ϕ1, . . . , ϕn) for a necessity measure µ and utility functions
ϕi if and only if there exist a utility function ϕ and a necessity
measure µ0, such that- is represented by f = qµ0(ϕ, . . . , ϕ).
To obtain function ϕ from ϕ1, . . . , ϕn, we can use ϕ(x) =
f (x, ..., x) =∨i∈Iµ({i}) ∧ ϕi(x) [9, 10].
5
CONCLUDING REMARKS
In the numerical setting, utility functions play a crucial role in the expressive power of the expected utility approach, introducing the subjective perception of (real-valued) consequences of acts and ex-pressing the attitude of the decision-maker in the face of uncertainty. In the qualitative and finite setting, the latter point is taken into ac-count by the choice of the monotonic set-function in the Sugeno inte-gral expression. So one might have thought that a direct appreciation of consequences is enough to describe a large class of preference re-lations. This paper questions this claim by showing that even in the fi-nite qualitative setting, the use of local utility functions increases the expressive power of Sugeno integrals, thus proving that the frame-work of qualitative MCDM is formally more general that the one of state-independent qualitative DMU. However , the fact that MCDM and DMU have the same expressive power when possibility and ne-cessity measures are used should facilitate the transposition of the possibilistic logic counterpart of qualitative DMU [19] to MCDM.
ACKNOWLEDGEMENTS
The first author is supported by the internal research project F1R-MTH-PUL-12RDO2 of the University of Lux-embourg. The third author acknowledges support by the T ´AMOP-4.2.1/B-09/1/KONV-2010-0005 program of the Na-tional Development Agency of Hungary, by the Hungarian NaNa-tional Foundation for Scientific Research under grants no. K77409 and K83219, by the National Research Fund of Luxembourg, and co-funded under the Marie Curie Actions of the European Commission (FP7-COFUND).
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